| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 6·9-s − 14-s + 16-s − 6·18-s + 25-s − 28-s + 12·29-s + 16·31-s + 32-s − 6·36-s − 20·37-s + 16·47-s + 49-s + 50-s − 4·53-s − 56-s + 12·58-s − 16·59-s + 16·62-s + 6·63-s + 64-s − 6·72-s − 20·74-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 2·9-s − 0.267·14-s + 1/4·16-s − 1.41·18-s + 1/5·25-s − 0.188·28-s + 2.22·29-s + 2.87·31-s + 0.176·32-s − 36-s − 3.28·37-s + 2.33·47-s + 1/7·49-s + 0.141·50-s − 0.549·53-s − 0.133·56-s + 1.57·58-s − 2.08·59-s + 2.03·62-s + 0.755·63-s + 1/8·64-s − 0.707·72-s − 2.32·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 274400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 274400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.106196972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.106196972\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.977646092372605556420891081568, −8.400283535029584108147575940934, −8.058922554914412122379413087558, −7.45420248729985818641828193491, −6.78248070030119110364171611800, −6.28136292817627051453406742773, −6.17616144092315208416665412995, −5.47254939185735647334240808646, −4.90054686617114529340998403638, −4.63923341949934829303628265619, −3.72179923773102648718320811330, −3.00381372509091040158022668795, −2.92257849808604956447035538813, −2.09663760025842118849257213691, −0.75960958349138602912815032456,
0.75960958349138602912815032456, 2.09663760025842118849257213691, 2.92257849808604956447035538813, 3.00381372509091040158022668795, 3.72179923773102648718320811330, 4.63923341949934829303628265619, 4.90054686617114529340998403638, 5.47254939185735647334240808646, 6.17616144092315208416665412995, 6.28136292817627051453406742773, 6.78248070030119110364171611800, 7.45420248729985818641828193491, 8.058922554914412122379413087558, 8.400283535029584108147575940934, 8.977646092372605556420891081568
